43 research outputs found
Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model
We derive the local statistics of the canonical ensemble of free fermions in
a quadratic potential well at finite temperature, as the particle number
approaches infinity. This free fermion model is equivalent to a random matrix
model proposed by Moshe, Neuberger and Shapiro. Limiting behaviors obtained
before for the grand canonical ensemble are observed in the canonical ensemble:
We have at the edge the phase transition from the Tracy--Widom distribution to
the Gumbel distribution via the Kardar-Parisi-Zhang (KPZ) crossover
distribution, and in the bulk the phase transition from the sine point process
to the Poisson point process. A similarity between this model and a class of
models in the KPZ universality class is explained. We also derive the
multi-time correlation functions and the multi-time gap probability formulas
for the free fermions along the imaginary time.Comment: 46 pages, 2 figure
Nonintersecting Brownian motions on the unit circle
We consider an ensemble of nonintersecting Brownian particles on the unit
circle with diffusion parameter , which are conditioned to begin at
the same point and to return to that point after time , but otherwise not to
intersect. There is a critical value of which separates the subcritical
case, in which it is vanishingly unlikely that the particles wrap around the
circle, and the supercritical case, in which particles may wrap around the
circle. In this paper, we show that in the subcritical and critical cases the
probability that the total winding number is zero is almost surely 1 as
, and in the supercritical case that the distribution of the total
winding number converges to the discrete normal distribution. We also give a
streamlined approach to identifying the Pearcey and tacnode processes in
scaling limits. The formula of the tacnode correlation kernel is new and
involves a solution to a Lax system for the Painlev\'{e} II equation of size 2
2. The proofs are based on the determinantal structure of the
ensemble, asymptotic results for the related system of discrete Gaussian
orthogonal polynomials, and a formulation of the correlation kernel in terms of
a double contour integral.Comment: Published at http://dx.doi.org/10.1214/14-AOP998 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Riemann-Hilbert Approach to the Six-Vertex Model
The six-vertex model, or the square ice model, with domain wall boundary
conditions (DWBC) has been introduced and solved for finite by Korepin and
Izergin. The solution is based on the Yang-Baxter equations and it represents
the free energy in terms of an Hankel determinant. Paul Zinn-Justin
observed that the Izergin-Korepin formula can be re-expressed in terms of the
partition function of a random matrix model with a nonpolynomial interaction.
We use this observation to obtain the large asymptotics of the six-vertex
model with DWBC. The solution is based on the Riemann-Hilbert approach. In this
paper we review asymptotic results obtained in different regions of the phase
diagram.Comment: 15 pages, 4 figures, Submitted to the MSRI volume on "Random Matrix
Theory, Interacting Particle Systems and Integrable Systems". arXiv admin
note: text overlap with arXiv:math-ph/051003
Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase
This is a continuation of the paper [4] of Bleher and Fokin, in which the
large asymptotics is obtained for the partition function of the
six-vertex model with domain wall boundary conditions in the disordered phase.
In the present paper we obtain the large asymptotics of in the
ferroelectric phase. We prove that for any \ep>0, as ,
Z_n=CG^nF^{n^2}[1+O(e^{-n^{1-\ep}})], and we find the exact value of the
constants and . The proof is based on the large asymptotics for
the underlying discrete orthogonal polynomials and on the Toda equation for the
tau-function.Comment: 22 pages, 7 figure
Uniform Asymptotics for Discrete Orthogonal Polynomials with Respect to Varying Exponential Weights on a Regular Infinite Lattice
We consider the large- asymptotics of a system of discrete orthogonal
polynomials on an infinite regular lattice of mesh , with weight
, where is a real analytic function with sufficient growth
at infinity. The proof is based on formulation of an interpolation problem for
discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert
problem, and steepest descent analysis of this Riemann-Hilbert problem.Comment: 32 pages, 4 figures; corrected versio
Two Lax systems for the Painlev\'e II equation, and two related kernels in random matrix theory
We consider two Lax systems for the homogeneous Painlev\'{e} II equation: one
of size studied by Flaschka and Newell in the early 1980's, and one
of size introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in
the early 2010's. We prove that solutions to the system can be
derived from those to the system via an integral transform, and
consequently relate the Stokes multipliers for the two systems. As corollaries
we are able to express two kernels for determinantal processes as contour
integrals involving the Flaschka-Newell Lax system: the tacnode kernel arising
in models of nonintersecting paths, and a critical kernel arising in a
two-matrix model.Comment: 46 pages, 20 figure
Six-vertex model with partial domain wall boundary conditions: ferroelectric phase
We obtain an asymptotic formula for the partition function of the six-vertex
model with partial domain wall boundary conditions in the ferroelectric phase
region. The proof is based on a formula for the partition function involving
the determinant of a matrix of mixed Vandermonde/Hankel type. This determinant
can be expressed in terms of a system of discrete orthogonal polynomials, which
can then be evaluated asymptotically by comparison with the Meixner
polynomials.Comment: 32 pages, 6 figures, minor changes in version
Nonintersecting Brownian bridges on the unit circle with drift
Nonintersecting Brownian bridges on the unit circle form a determinantal
stochastic process exhibiting random matrix statistics for large numbers of
walkers. We investigate the effect of adding a drift term to walkers on the
circle conditioned to start and end at the same position. For each return time
we show that if the absolute value of the drift is less than a
critical value then the expected total winding number is asymptotically zero.
In addition, we compute the asymptotic distribution of total winding numbers in
the double-scaling regime in which the expected total winding is finite. The
method of proof is Riemann--Hilbert analysis of a certain family of discrete
orthogonal polynomials with varying complex exponential weights. This is the
first asymptotic analysis of such a class of polynomials. We determine
asymptotic formulas and demonstrate the emergence of a second band of zeros by
a mechanism not previously seen for discrete orthogonal polynomials with real
weights.Comment: 40 pages, 11 figure
The k-tacnode process
The tacnode process is a universal behavior arising in nonintersecting
particle systems and tiling problems. For Dyson Brownian bridges, the tacnode
process describes the grazing collision of two packets of walkers. We consider
such a Dyson sea on the unit circle with drift. For any integer k, we show that
an appropriate double scaling of the drift and return time leads to a
generalization of the tacnode process in which k particles are expected to wrap
around the circle. We derive winding number probabilities and an expression for
the correlation kernel in terms of functions related to the generalized
Hastings-McLeod solutions to the inhomogeneous Painleve-II equation. The method
of proof is asymptotic analysis of discrete orthogonal polynomials with a
complex weight.Comment: 38 pages, 8 figure
The Fourier extension method and discrete orthogonal polynomials on an arc of the circle
The Fourier extension method, also known as the Fourier continuation method,
is a method for approximating non-periodic functions on an interval using
truncated Fourier series with period larger than the interval on which the
function is defined. When the function being approximated is known at only
finitely many points, the approximation is constructed as a projection based on
this discrete set of points. In this paper we address the issue of estimating
the absolute error in the approximation. The error can be expressed in terms of
a system of discrete orthogonal polynomials on an arc of the unit circle, and
these polynomials are then evaluated asymptotically using Riemann--Hilbert
methods.Comment: 47 pages, 3 figure